Mathematical Principles of Theoretical Physics

2.5. PRINCIPLE OF LAGRANGIAN DYNAMICS (PLD) 85

By (2.5.23), we have

∇L=

e

c

∇~A·v+e∇A 0.

Thanks to

∇~A·v=∇(~A·v) (by∂v= 0 )

=(~A·∇)v+ (v·∇)~A+v×curl~A+~Acurlv

=(v·∇)~A+v×curl~A,

the equation (2.5.25) reads

(2.5.26)

dP

dt

+

e

c

d~A

dt

=

e

c

(v·∇)~A+

e

c

v×curl~A+e∇A 0.

It is known that
d~A
dt

=

∂~A

∂t

+

∂~A

∂xk

dxk

dt

=

∂~A

∂t

+ (v·∇)~A.

Hence (2.5.26) is rewritten as

(2.5.27)
dP
dt

=−

e

c

∂~A

∂t

+e∇A 0 +

e

c

v×curl~A.

Then, by (2.5.20), the equation (2.5.27) takes the form

dP

dt

=eE+

e

c

v×curl~A.

Thus, we have deduced (2.5.21) from the action (2.5.23).
Now, we deduce the Einstein energy-momentum formula for the4-dimensional energy-
momentum under an electromagnetic field. Corresponding to the least action principle of
classical mechanics, the momentumPand energyEof a charged particle are given by

(2.5.28) P=

∂L

∂v

, E=v·

∂L

∂v

−L.

By (2.5.23), we refer from (2.5.28) that

(2.5.29)

P=

mv

√

1 −v^2 /c^2

+

e

c

~A, ~Athe magnetic potential,

E=

mc^2

√

1 −v^2 /c^2

+eA 0 , A 0 the electric potential.

It follows from (2.5.29) that

(2.5.30) (E−eA 0 )^2 =c^2 (~P−
e
c

~A)^2 +m^2 c^4 ,

which is the Einstein energy-momentum relation under with an electromagnetic field.
It is the formula (2.5.30) that makes us to take the energy and momentum operators in
quantum mechanics in the following form:

(2.5.31) ih ̄

∂

∂t

−eA 0 , −ih ̄∇−

c

e

~A,

which were also given in (2.2.52). In particular, the particular form of (2.5.31) leads to the
origin of gauge theory.